Abstract
We investigate the infinite volume limit of quantized photon fields in multimode coherent states. We show that for states containing a continuum of coherent modes, it is mathematically and physically natural to consider their phases to be random and identically distributed. The infinite volume states give rise to Hilbert space representations of the canonical commutation relations which we construct concretely. In the case of random phases, the representations are random as well and can be expressed with the help of Itô stochastic integrals. We analyze the dynamics of the infinite state alone and the open system dynamics of small systems coupled to it. We show that under the free field dynamics, initial phase distributions are driven to the uniform distribution. We demonstrate that coherences in small quantum systems, interacting with the infinite coherent state, exhibit Gaussian time decay. The decoherence is qualitatively faster than the one caused by infinite thermal states, which is known to be exponentially rapid only. This emphasizes the classical character of coherent states.
Similar content being viewed by others
References
Araki H., Woods E.: Representations of canonical commutation relations describing a non-relativistic infinite free Bose gas. J. Math. Phys. 4, 637–662 (1963)
Araki H., Wyss W.: Representations of canonical anticommutation relations. Helv. Phys. Acta 37, 136–159 (1964)
Billingsley P.: Probability and Measure. Wiley, New York (1995)
Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics 1 and 2. Springer, New York (1987)
Breuer H.-P., Petruccione F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2006)
Coish W.A., Fischer J., Loss D.: Exponential decay in a spin bath. Phys. Rev. B 77, 125329 (2000)
de Sousa R., Das Sarma S.: Theory of nuclear-induced spectral diffusion: spin decoherence of phosphorus donors in Si and GaAs quantum dots. Phys. Rev. B 68, 115322 (2003)
Fröhlich J., Merkli M.: Thermal Ionization. Math. Phys. Anal. Geom. 7(3), 239–287 (2004)
Fujii, K.: Introduction to Coherent States and Quantum Information Theory. arXiv:quant-ph/0112090v2
Fujii K., Oike H.: Basic properties of coherent-squeezed states revisited. Int. J. Geom. Methods Mod. Phys. 11(5), 1450051, 15 (2014)
Gardiner, C.W., Zoller, P.: Quantum Noise. Springer Series in Synergetics, 3rd edn (2004)
Glauber R.J.: The quantum theory of optical coherence. Phys. Rev. 130, 2529–2539 (1963)
Glauber R.J.: Coherent and incoherent states of the radiation field. Phys. Rev. 131, 2766–2788 (1963)
Grosshans F., Van Assche G., Wenger J., Brouri R., Cerg N.J., Ph. Grangier: Quantum key distribution using gaussian-modulated coherent states. Lett. Nat. Nat. 421, 238–241 (2003)
Hirvensalo M.: Quantum Computing. Springer Verlag (Natural computing series), New York (1998)
Joos E., Zeh H.D., Kiefer C., Giulini D., Kupsch J., I.O. Stamatescu: Decoherence and the Appearence of a Classical World in Quantum Theory, 2nd edn. Springer, Berlin (2003)
Kaye P., Laflamme R., Mosca M.: An introduction to Quantum Computing. Oxford University Press, Oxford (2007)
Klauder J.R., Skagerstam B.-S.: Coherent States, Applications in Physics and Mathematical Physics. World Scientific, Singapore (1985)
Mandel L., Wolf E.: Optical Coherence and Quantum Optics. Cambridge University Press, Cambridge (1995)
Martin Ph.A., Rothen F.: Many-Body Problems and Quantum Field Theory. Springer Texts and Monographs in Physics. Springer, New York (2004)
Mosheni M., Omar Y., Engel G.S., Plenio M.B.: Quantum Effects in Biology. Cambridge University Press, Cambridge (2014)
Merkli, M., Berman, G.P., Sayre, R.T., Gnanakaran, S., Könenberg, M., Nesterov, A.I., Song, H.: Dynamics of a Chlorophyll Dimer in Collective and Local Thermal Environments. J.Math. Chem. 54(4), 866–917 (2016)
Merkli M., Berman G.P., Sayre R.: Electron transfer reactions: generalized Spin-Boson approach. J. Math. Chem. 51(3), 890–913 (2013)
Merkli M., Sigal I.M., Berman G.P.: Resonance theory of decoherence and thermalization. Ann. Phys. 323, 373–412 (2008)
Merkli, M.: The ideal quantum gas. In: Attal, S., Joye, A., Pillet, C.-A (eds.) Lecture Notes in Mathematics 1880. Springer, New York (2006)
Moncrief V.: Coherent states and quantum nonperturbing measurements. Ann. Phys. 114, 1–2 (1978)
Myatt C.J., King B.E., Turchette Q.A., Sackett C.A., D. Kielpinski, Itano W.M., Monroe C., Wineland D.J.: Decoherence of quantum superpositions through coupling to engineered reservoirs. Nature 403, 269–273 (2000)
Nakahara M., Ohmi T.: Quantum Computing, from Linear Algebra to Physical Realizations. CRC Press, Boca Raton (2008)
Nielsen M.A., Chuang I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Øksendal, B.: Stochastic Differential Equations, 6th edn. Universitext. Springer, New York (2003)
Poyatos J.F., Cirac J.I., Zoller P.: Quantum reservoir engineering with laser cooled trapped ions. Phys. Rev. Lett. 77, 23 (1996)
Palma M.G., Suominen K.-A., Ekert A.: Quantum computers and dissipation. Proc. R. Soc. Lond. Ser. A 452, 567–584 (1996)
Schrödinger E.: Der stetige Übergang von der Mikro- zur Makromechanik. Naturwissenschaften 14, 664–666 (1926)
Schlosshauer M.: Decoherence and the Quantum-to-Classical Transition. The Frontiers Collection. Springer, New York (2007)
Xu D., Schulten K.: Coupling of protein motion to electron transfer in a photosynthetic reaction center: investigating the low temperature behavior in the framework of the spin boson model. Chem. Phys. 182, 91–117 (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
Rights and permissions
About this article
Cite this article
Joye, A., Merkli, M. Representations of Canonical Commutation Relations Describing Infinite Coherent States. Commun. Math. Phys. 347, 421–448 (2016). https://doi.org/10.1007/s00220-016-2611-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-016-2611-1